Definition 6.3.1 (Pseudometric).label Let $X$ be a set, and $d: X \times X \to [0, \infty)$, then $d$ is a pseudometric on $X$ if

1. (PM1)

   For any $x \in X$, $d(x, x) = 0$.

2. (PM2)

   For any $x, y \in X$, $d(x, y) = d(y, x)$.

3. (PM3)

   For any $x, y, z \in X$, $d(x, z) \le d(x, y) + d(y, z)$.

If $d$ satisfies the above and

1. (M)

   For any $x, y \in X$ with $x \ne y$, $d(x, y) > 0$.

then $d$ is a metric.